Every Syzygy is a linear combination of pair-wise Syzygies

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Im working on understanding Gröbner bases. I've understood how to show existance and uniqueness(of reduced Gröbner bases).
To understand how to actually compute them, I need to understand Syzygies in free modules.
The theorem reads thus:
In a ring of multivariate polynomials over a field, if S =(s_1,s_2,s_3...s_n) is a syzygy of (m_1,m_2,m_3...m_n), where every m_i is a monomial, S is a linear combination of the canonical pair-wise Syzygies.

I've been trying to get some headway on this proof for a week now, with little success.
Any comments or hints appreciated! Thank you!